Flexible domains for minimal surfaces in Euclidean spaces

TL;DR

This paper introduces a new concept of flexibility for domains in Euclidean spaces based on the approximation of conformal minimal immersions, extending the flexibility-rigidity dichotomy from complex analysis to minimal surface theory.

Contribution

It defines and studies the notion of flexible domains in Euclidean spaces, broadening the understanding of minimal surface embeddings and their approximation properties.

Findings

Flexible domains allow uniform approximation of minimal immersions.

The work extends the flexibility-rigidity dichotomy to minimal surface theory.

It connects hyperbolicity phenomena with domain flexibility in Euclidean spaces.

Abstract

In this paper we introduce and investigate a new notion of flexibility for domains in Euclidean spaces $\mathbb R^n$ for $n\ge 3$ in terms of minimal surfaces which they contain. A domain $\Omega$ in $\mathbb R^n$ is said to be flexible if every conformal minimal immersion $U\to\Omega$ from a Runge domain $U$ in an open conformal surface $M$ can be approximated uniformly on compacts, with interpolation on any given finite set, by conformal minimal immersion $M\to \Omega$. Together with hyperbolicity phenomena considered in recent works, this extends the dichotomy between flexibility and rigidity from complex analysis to minimal surface theory.